In 1992 Escobar [1]} generalized the classical Yamabe problem
to compact Riemannian manifolds with regular boundary. In this case
one can ask if there exists a conformal metric \(\tilde{g}\)  which
has both constant scalar curvature and constant mean curvature of
the boundary. In this case the analytic formulation for the Yamabe
problem takes the form
\(\left\lbrace  \begin{array}{ll}-\Delta _{g}u+\frac{n-2}{4(n-1)}R_{\varepsilon }u=c_{1}u^{\frac{n+2}{n-2}} & \text{ on }M\\\frac{\partial }{\partial \nu }u+\frac{n-2}{2}h_{g}(x)u=c_{2}u^{\frac{n}{n-2}} & \text{ on }\partial M\end{array}\right..\) 
